brainwagon

As I might have mentioned, I am trying to teach myself a bit about electronics and radio design. I find the problem with being self taught is that often you read something, and it doesn’t seem clear to you why it should be so, and you uncover the basic lack of understanding that you have to go back and fill in before the later material makes sense. This happened to me while I was trying to work through some more amplifier designs, and I realized that I had a basic misconception about how impedance matching actually worked, and this was perhaps even more basic, and could be illustrated with a simple circuit, consisting of a voltage source, and then in series a source resistance R_S and a load resistance R_L. The question is given an R_S, what value of R_L maximizes the power dissapated in the load R_L?

Well, it’s not really too hard to figure out. First, we can determine the total current going around the loop. For a voltage V, the current is simply:

       V
I = -------
    R  + R
     S    L

From this we can easily determine the voltage drop across R_L:

      R  V
       L
V  = -------
 L   R  + R
      S    L

And, since we know the current through and the voltage across the resistor, we can determine the power as their product:

            2
        R  V
         L
P  = ----------
 L            2
     /R  + R \
     \ S    L/

When is power maximized? Well, we can differentiate the power equation with respect to R_L, and we get:

                           2
dP         2         2 R  V
  L       V             L
--- = ---------- - ----------
dR             2            3
  L   /R  + R \    /R  + R \
      \ S    L/    \ S    L/

Setting this equal to zero, and solving for R_L, we find that a maximum occurs where R_L equals R_S, in other words, power is maximized when the source and load resistances are matched. The total disappated by R_L is then:

      2
     V
P  = ---
 L   4 R

Thus, if we had a load of 50 ohms and a voltage source of 12 volts, we’d end up with a maximum power of 720 mw.

Addendum: As a double check, the power passing through both resistors is V^2 / 2R, which would have been 1.44 watts, and obviously since both resistors are the same, the power is evenly split between the two.

Addendum²: I forgot to mention what I was confused about. It’s not really this (which occurs in DC circuits) but the corresponding circuit which occur in AC circuits with complex impedances. I’ll work through this later.

Addendum³: Hmm. Subscripts and superscripts seem to not work right with this theme. I’ll fix it.